To fully expand [tex]\((m+2)^4\)[/tex], let's simplify each term step-by-step:
1. First Term:
[tex]\[(1) \cdot m^4 \cdot 2^0 = 1 \cdot m^4 \cdot 1 = m^4\][/tex]
2. Second Term:
[tex]\[(4) \cdot m^3 \cdot 2^1 = 4 \cdot m^3 \cdot 2 = 8m^3\][/tex]
3. Third Term:
[tex]\[(6) \cdot m^2 \cdot 2^2 = 6 \cdot m^2 \cdot 4 = 24m^2\][/tex]
4. Fourth Term:
[tex]\[(4) \cdot m \cdot 2^3 = 4 \cdot m \cdot 8 = 32m\][/tex]
5. Fifth Term:
[tex]\[(1) \cdot 2^4 = 1 \cdot 16 = 16\][/tex]
Putting it all together, we have:
[tex]\[
(m+2)^4 = m^4 + 8m^3 + 24m^2 + 32m + 16
\][/tex]
So the full expansion is:
[tex]\[
(m+2)^4 = m^4 + 8m^3 + 24m^2 + 32m + 16
\][/tex]
Therefore, the terms you need to fill in are:
[tex]\[
m^4 + \boxed{8m^3} + \boxed{24m^2} + \boxed{32m} + \boxed{16}
\][/tex]
